Does Blender remember every manipulation done?
No it doesn't remember edits for playback like you want (and like ZBrush and others can).

What you would need to do is to export each step as an .obj, load them all into a new scene and animate them so they appear in a sequence one after another.

Then you can render this with orbiting camera and with wire-frame modifier on.

Using scripting to do this would be smart, there are many answers here how to import multiple obj's and how to animate multiple objects.

Some addons can store geometry states into a buffer, but there is no way that I know of how to efficiently animate them.